VCE Operation in Function Relation Composition (f+g)(x), (f-g)(x) ,(fg)x ,(f/g)x and (f.g)x

Concepts

Algebraic combinations of functions, composition, and decomposition of functions.

Algebraic Combinations of Functions

An ambitious way of creating new functions is to combine two or more functions to create a new function. The most obvious way we can do this is to perform basic algebraic operations on the two functions to create the new one; hence we can add, subtract, multiply, or divide functions.

Note that there are two types of algebras in use in this section:

• The algebra of real numbers, e.g., 4 × 5 = 20, 4 − 5 = −1, 20/10 = 2, etc.
• The algebra of functions, e.g., f + g, f − g, etc.

Algebra of Functions

Let f (with domain A) and g (with domain B) be functions. Then the functions f + g, f − g, fg, f / g are defined as:

       (f + g)(x) = f(x) + g(x), domain: A ∩ B
       (f − g)(x) = f(x) − g(x), domain: A ∩ B
          (fg)(x) = f(x) * g(x), domain: A ∩ B
         (f/g)(x) = f(x) / g(x), domain: {x ∈ A ∩ B | g(x) ≠ 0}
        

The domains are the intersection of the domains of f and g, ensuring that division by zero does not occur.

A Closer Look

• The minus sign in f − g represents the difference between two functions.
• The minus sign in f(x) − g(x) represents the difference between two real numbers.

The relation (f − g)(x) = f(x) − g(x) allows us to calculate this quantity, which is easy to remember. Understanding mathematical notation is key.

Note: Two functions are equal if they have the same functional definition and the same domain.

Example

If f(x) = √x and g(x) = √(4 − x²), find f + g, f − g, fg, f / g and their domains.

Domain of f = [0, ∞)
Domain of g = [−2, 2]
Intersection: [0, 2]

(f + g)(x) = √x + √(4 − x²), 0 ≤ x ≤ 2
(f − g)(x) = √x − √(4 − x²), 0 ≤ x ≤ 2
   (fg)(x) = √x √(4 − x²)  , 0 ≤ x ≤ 2
  (f/g)(x) = √x / √(4 − x²), 0 ≤ x < 2
        

Important Note:

For (f/g)(x) we exclude x = 2 since it would lead to division by zero. Divide by Zero is undefined.

Composition of Functions

Given two functions f and g, the composite function f ◦ g is defined as:

(f ◦ g)(x) = f(g(x))

The domain of f ◦ g includes all x-values in the domain of g that map to values of g(x) in the domain of f. Note that f ◦ g ≠ g ◦ f.

Example

If f(x) = √x and g(x) = √(4 − x²), find f ◦ g and g ◦ f and their domains.

(f ◦ g)(x) = √(√(4 − x²))
Domain: [−2, 2]

(g ◦ f)(x) = √(4 − √x²)
Domain: [0, 2]
        

Important Note:

It is important to note that f ◦ g ≠ g ◦ f.